International Journal of Scientific Research and Engineering Development— Volume 4 Issue 1, Jan-Feb 2021 Available at www.ijsred.com

RESEARCH ARTICLE OPEN ACCESS

BASICS OF QUANTUM MECHANICAL SPIN AND Its CALCULATIONS Nischal Khanal *(SEDS sxc, St. Xavier’s College, Kathmandu)

************************ Abstract: In this article classical understanding of spin has been exploited to model and visualize the concept of quantum mechanical spin. Quantum mechanical spin holds significance in understanding the system of both the macro and quantum worlds. This article explains the existence of spin and its mechanism in particles studying the results obtained by Stern and Gerlach in 1922 AD through their experiment. The article explicitly explains the connection between spin and magnetism. Moreover, the study was made to ease the concept of Quantum mechanical spin for general understanding for readers and enthusiasts. Furthermore, the introductory calculations of spin of an electron will enlighten the readers towards the mathematical understanding of spin and the drawbacks of some of the proposed electron spin models that evaluate the value of angular momentum and magnetic moment.

Keywords —Magnetic moment, Angular momentum, spin , Wave function, Inhomogeneousmagnetic field, Orbital, Bloch sphere, charge density

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I. INTRODUCTION II. BACKGROUND ( ORIGIN OF EXISTENCE ) The classical mechanics have always described Spin was first discovered in the context of the the spin as an Angular momentum of macro emission spectrum of alkali metals. In 1924, objects. As compared to Quantum mechanics, Spin Wolfgang Pauli introduced what he called a "two- has bizarre definition and understanding in the valuedness not describable classically" associated quantum world. In quantum mechanics and particle with the electron in the outermost shell. This physics, spin is an intrinsic form of angular allowed him to formulate the Pauli Exclusion momentum carried by elementary particles, Principle, stating that no two electrons can have the composite particles ( hadrons ), and atomic nuclei. same quantum state in the same quantum system. Spin is one of two types of angular momentum in The physical interpretation of Pauli's "degree of quantum mechanics, the other being orbital angular freedom" was initially unknown. Ralph Kronig, one momentum. The orbital angular momentum of Landé's assistants, suggested in early 1925 that it operator is the quantum-mechanical counterpart to was produced by the self-rotation of the electron. the classical angular momentum of the orbital When Pauli heard about the idea, he criticized it revolution and appears when there is a periodic severely, noting that the electron's hypothetical structure to its wave function as the angle surface would have to be moving faster than the varies.[ 1] In practice, Spin is assigned a quantum speed of light in order for it to rotate quickly number i.e. division of spin enough to produce the necessary angular angular momentum by reduced Planck constant( ℏ). momentum. This would violate the theory of Particles with half-integer spin ( 1/2 ℏ, 3/2 ℏ,5/2 ℏ, relativity. Largely due to Pauli's criticism, Kronig decided not to publish his idea. [ 2] etc .) are called fermions. Particles with integer spin ℏ, ℏ, ℏ, ( 1 2 3 etc. ) are called bosons.[ 2]

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International Journal of Scientific Research and Engineering Development— Volume 4 Issue 1, Jan-Feb 2021 Available at www.ijsred.com In retrospect, the genesis of this bizarre spin The reason is that the magnetic fields generated by mechanics dates back to 1922 when Stern and individual particles (mostly electrons, whose Gerlach delivered the first experimental proof of the magnetic fields are much stronger than those of fascinating degree of freedom of an electron spin. protons or neutrons) often cancel out, which in turn In the Stern-Gerlach experiment, a beam of silver makes most materials non-magnetic. For instance, if atom is generated in an atomic beam furnace, it was an atomic orbital is completely filled, electrons in then sent towards an inhomogeneous magnetic this orbital have opposite spins that cancel out and field. According to classical physics, the results hence, show no magnetic behavior. This means that would have been expected for the magnetic no atom with filled or almost filled orbitals can be moments of silver atoms to be randomly oriented so magnetic. In order to possess magnetic properties, that they should be deflected in the inhomogeneous an atom should have half-filled orbitals. So that the magnetic field by different amounts depending on magnetic fields of individual electrons reinforce one their orientation. However, the results of the another. researchers observed that the beam was split into two possible states which were later named spin up and spin down. If we look at the configuration of silver (Ag) atom then we can see the each silver atom has total of 47 electrons in which 46 of the electrons each spin up is paired with one spin down, spins neutralize each other and cancel out the each other but, still the unpaired spin either spin up or spin down state resulting in an anomalous magnetic behavior of an electron. We can represent the spin state using a bloch sphere where the spin can point in any random direction, similarly, the spin of silver atoms are randomly distributed. Therefore, the silver atoms are not polarized. Fig. 1.1 http://stonebrakerdesignworks.com

However, not all materials made up of magnetic III. SPIN MAGNETIC MOMENT atoms exhibit magnetic properties. This due to the A spin magnetic moment is a magnetic moment configuration of individual atoms. Many materials caused by the spin of elementary particles. For have their atoms arranged so that their magnetic example, the electron is an elementary spin-1/2 fields cancel out. Only a fraction of materials have fermion. Quantum electrodynamics gives the most the atoms accurate prediction of the anomalous magnetic Arranged so that their magnetic fields mutually moment of the electron. [ 3] In light of the reinforce. This is why magnetic materials are so connection between spin and magnetic moment, It rare. The comparison of particles to tiny magnets is turns out particles behave like peculiar tiny magnets not accurately true. It’s because the magnetic fields by generating weak magnetic fields. That is why created by particles behave rather oddly. We can objects from macro world, which are composed of demonstrate this in a simple experiment. Say we many such “tiny magnets” are magnetic. But that have two axes: x and y (i.e. if we were to measure does not explain why only a handful of materials its spin in the direction of the x-axis, we would get are magnetic when all macroscopic materials are 1). But what happens if we try to measure its made up of these tiny magnets. How is that spin(magnetic field) on the y-axis? If we took a classical magnet, pointed it in the direction x, and possible?

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International Journal of Scientific Research and Engineering Development— Volume 4 Issue 1, Jan-Feb 2021 Available at www.ijsred.com

Conducted the same experiment, we would measure Based on external motion equations of the electrons no magnetic field pointing in the y-direction, of could only give the valid value of angular course (since the x-axis is perpendicular to the y- momentum and magnetic motion. As of now, axis).However, particles behave in a completely there's no unified model that can both elucidate the different way. If we measure the spin of a particle value of angular momentum and magnetic moment in the y-direction, half the time we get to spin 1, the and the mass origin. Some of the proposed models other half of the time we get -1. Even if we try to have failed to give the correct value of angular measure the spin in different directions, we always momentum and magnetic moment. They could not get either 1 or -1. However, the average of the even explain the forces that bind the electric values we get is always equal to the value we would charges together in the electron. expect to get with classical magnets. We can To describe the electron spin motion at the demonstrate this rule on our particle. The average quantum mechanics level, we must get a spin wave of the values we got (half the time 1, half the time - function, just as the orbital wave function for the 1) Is equal to zero, which is the value we would get motion of the electron around the nucleus.[] To with a normal magnet? calculate the spin angular momentum and magnetic IV.SPIN CALCULATIONS AT QUANTUM moment, we first make some assumptions for the SCALE electron structure. Enlightenment can be achieved from the mass distribution of quarks. Particle physics tells us that the mass of up/down current In Quantum mechanics, we deal with the quark (bare quark) is only several Mev, while the calculations different from the problems we see in quark's constituent mass is more than 300Mev. so classical electron spin models. The mass origin and most of the quark mass comes from the effective the radius of the electron are long-standing masses of the sea quarks (virtual quark-antiquark problems in classical and modern physics. If the pairs) and the virtual gluons. Similarly, we think electron's mass is totally due to electromagnetic that the electron is composed of point electric origin, then the classical radius of the electron will charge at the center, virtual electron-positron pairs be the order of 10-15 m. However, if we wish to get inside the electron and virtual photons both inside the spin value based on that hypothesis, the electron and outside the electron. [ 5] surface’s rotational speed would be more than 100 c. [ 4], which seems unrealistic and unreasonable. The spin of an electron owes a relativistic effect as ACKNOWLEDGMENT it is derived from the Dirac equation. Often I would like to thank the entire team of SEDS sxc electrons are regarded as the point particles in and ecstatic paradox, who helped me refine this quantum field theory. However, if this has to be article for general understating. true, certain rotational effects would produce spin in an electron. It explicitly shows that the generated REFERENCES angular momentum and magnetic moment of an electron would always obey the classical relations. [1] Merzbacher, Eugen (1998). Quantum Mechanics (3rd ed.). pp. 372 Griffiths, David (2005). Introduction to Quantum Mechanics (2nd Therefore, the electron cannot be a point particle if ed.).pp. 1 it has to follow the quantum relations. And should [2] Pais, Abraham (1991). Niels Bohr's Times . Oxford: Clarendon Press. pp. 201 . have varying mass and charge density. At present, [3] Dirac, P. A. M. (1 February 1928). "The Quantum Theory of the Electron" several proposed electron spin models suggest the [4] Zhang Y. D., Quantum mechanics, Science Press, Beijing, 2002. [5] Quantum Mechanical Calculation of Electron Spin Hai-Long Zhao1 values of angular momentum and magnetic moment [6] Scerri, Eric R. (1995). "The exclusion principle, chemistry and hidden variables". to be correct. Still, only the handful of models

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